The lattice copies of ℓ 1 in Banach lattices

Marek Wójtowicz

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 649-653
  • ISSN: 0010-2628

Abstract

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It is known that a Banach lattice with order continuous norm contains a copy of ℓ 1 if and only if it contains a lattice copy of ℓ 1 . The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the c 0 - and ℓ ∞ -cases considered by Lozanovskii, Mekler and Meyer-Nieberg.

How to cite

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Wójtowicz, Marek. "The lattice copies of $\ell _1$ in Banach lattices." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 649-653. <http://eudml.org/doc/248806>.

@article{Wójtowicz2001,
abstract = {It is known that a Banach lattice with order continuous norm contains a copy of $\ell _1$ if and only if it contains a lattice copy of $\ell _1$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the $c_0$- and $\ell _\{\infty \}$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.},
author = {Wójtowicz, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Banach lattice; order continuous norm; embedding of $\ell _1$; Banach lattice; order continuous norm; embedding of },
language = {eng},
number = {4},
pages = {649-653},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The lattice copies of $\ell _1$ in Banach lattices},
url = {http://eudml.org/doc/248806},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Wójtowicz, Marek
TI - The lattice copies of $\ell _1$ in Banach lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 649
EP - 653
AB - It is known that a Banach lattice with order continuous norm contains a copy of $\ell _1$ if and only if it contains a lattice copy of $\ell _1$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the $c_0$- and $\ell _{\infty }$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.
LA - eng
KW - Banach lattice; order continuous norm; embedding of $\ell _1$; Banach lattice; order continuous norm; embedding of
UR - http://eudml.org/doc/248806
ER -

References

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  9. Lindenstrauss J., Tzafriri L., Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977. Zbl0362.46013MR0500056
  10. Lindenstrauss J., Tzafriri L., Classical Banach Spaces II, Function Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1979. Zbl0403.46022MR0540367
  11. Lozanovskii G.Ya., On one result of Shimogaki (in Russian), Theses of Second Conference of the Pedagogical Institutes of Nord-West Region Devoted to Mathematics and Methods of its Teaching, Leningrad, 1970, 43. 
  12. Meyer-Nieberg P., Banach Lattices, Springer-Verlag, Berlin-Heidelberg-New York, 1991. Zbl0743.46015MR1128093
  13. de Pagter B., Wnuk W., Some remarks on Banach lattices with non-atomic duals, Indag. Math. (N.S.) 1 (1990), 391-395. (1990) Zbl0731.46008MR1075887
  14. Polyrakis I., Lattice-subspaces of C [ 0 , 1 ] and positive bases, J. Math. Anal. Appl. 184 (1994), 1-18. (1994) Zbl0802.46035MR1275938
  15. Wnuk W., Locally solid Riesz spaces not containing c 0 , Bull. Polish Acad. Sci. Math. 36 (1988), 51-55. (1988) MR0998207
  16. Wnuk W., Banach Lattices with Order Continuous Norm, Polish Scientific Publishers, Warszawa, 1999. 
  17. Wójtowicz M., The Sobczyk property and copies of ℓ ∞ in locally convex-solid Riesz spaces, Arch. Math. 75 (2000), 376-379. (2000) MR1785446

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